Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×M16 | C42 |
| Frattini, cosocle | C2×#407 | C22 |
| socle | C22 | #1343 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ C2×#407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
408 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(5,6)(7,17)(8,22)(9,14)(10,15)(11,16)(12,18)(13,24)(19,33)(20,29)(21,34)(23,31)(25,32)(26,30)(27,36)(28,35);(5,7,9,19,11,21,10,20)(6,12,14,27,16,28,15,31)(8,23,24,35,25,36,26,18)(13,29,22,17,30,33,32,34)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gh,e^c=ehi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1496);